motor - Does cassette tape change its velocity while playing?

Looking at the old cassette tape,

From the POV of the head, let's say that it reads at speed $v$ (the magnetic medium scrolls at speed $v$).

But looking at the right wheel, which is the one that's pooling the magnetic medium - its radius is growing(!) over time.

Now, $v=r\omega$, where $\omega$ is the angular velocity, i.e. a constant

Question

I don't think that's true. What is really going on here? Radius is growing over time, for sure. I also assume that $\omega$ is constant. so did $v$ increase?

Answer

The details of how a cassette drive works are well covered by this Wikipedia article. The tape is pulled by a capstan next to the playback head, and this capstan pulls the tape at a steady rate.

(picture from the Wikipedia article)

You probably need to click on the picture to see it full size. I have indicated the capstan by a red arrow. The take-up spool doesn't rotate at a fixed speed. It uses a slipping drive, as badjohn says in his answer, so it takes up the tape at the speed the capstan moves it.